Final answer:
To find QK, we can use the fact that the side lengths of the quadrilateral AXYZ are equal to XY, XZ, JQ, and QK. Using the formula for the area of a cyclic quadrilateral and the fact that the radius of the circumscribed circle of AXYZ is 17, we can find the length of QK to be 90.
Step-by-step explanation:
To find QK, we can use the fact that the side lengths of the quadrilateral AXYZ are equal to XY, XZ, JQ, and QK. Since XY = 30 and XZ = 24, we know that AY = 30 and AZ = 24. Additionally, since JQ = 8, we know that JA = JQ = 8. Using the fact that the radius of the circumscribed circle of AXYZ is 17, we can apply the formula for the area of a cyclic quadrilateral to find the length of QK.
The formula for the area of a cyclic quadrilateral is: Area = sqrt((s - AB)(s - BC)(s - CD)(s - DA)), where s is the semiperimeter and AB, BC, CD, and DA are the side lengths of the quadrilateral. In this case, AB = XY = 30, BC = AZ = 24, CD = XZ = 24, and DA = AY = 30.
Substituting the values, we have s = (AB + BC + CD + DA)/2 = (30 + 24 + 24 + 30)/2 = 54. Plugging the values into the formula, we get: Area = sqrt((54 - 30)(54 - 24)(54 - 24)(54 - 30)) = sqrt(24*30*30*24) = 720.
Now, using the formula for the area of a triangle (Area = 0.5 * base * height), we can find the height of triangle AQK. We know that the base AQ is 8 and the area is 720, so we can solve for the height: 720 = 0.5 * 8 * height, height = 720 / (0.5 * 8) = 90.
Therefore, we know that the height of triangle AQK is 90. Since QK is the base of this triangle, QK is also 90.